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Pullback attractors of reaction-diffusion inclusions with space-dependent delay
Topological stability in set-valued dynamics
1. | Instituto de Matemàtica y Ciencias Afines (IMCA), Universidad Nacional de Ingeniera Calle Los Biòlogos 245, 15012 Lima, Perù |
2. | Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530,21945-970 Rio de Janeiro, Brazil |
3. | Institut de Mathématiques Université de Bordeaux Ⅰ, 33405, Talence, France |
We propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the set-valued sense is topologically stable in the classical sense [
References:
[1] |
N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Mathematical Library, 52. North-Holland Publishing Co. , Amsterdam, 1994. |
[2] |
J. -P. Aubin and H. Frankowska, Set-valued Analysis, Reprint of the 1990 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc. , Boston, MA, 2009. |
[3] |
R. Bowen,
ω-limit sets for axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.
doi: 10.1016/0022-0396(75)90065-0. |
[4] |
D. Carrasco-Olivera, A. R. Metzger and C. A. Morales, Logarithmic expansion, entropy and dimension for set-valued maps, Preprint, (2016), to appear. |
[5] |
D. Carrasco-Olivera, R. Metzger Alvan and C. A. Morales,
Topological entropy for set-valued maps, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3461-3474.
doi: 10.3934/dcdsb.2015.20.3461. |
[6] |
W. Cordeiro and M. J. Pacifico,
Continuum-wise expansiveness and specification for set-valued functions and topological entropy, Proc. Amer. Math. Soc., 144 (2016), 4261-4271.
doi: 10.1090/proc/13168. |
[7] |
M. Eisenberg,
Expansive transformation semigroups of endomorphisms, Fund. Math., 59 (1966), 313-321.
|
[8] |
J. P. Kelly and T. Tennant, Topological entropy for set-valued functions, arXiv: 1509.08413. |
[9] |
S. Y. Pilyugin and J. Rieger,
Shadowing and inverse shadowing in set-valued dynamical systems. Contractive case, Topol. Methods Nonlinear Anal., 32 (2008), 139-149.
|
[10] |
S. Y. Pilyugin,
Shadowing in Dynamical Systems Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999. |
[11] |
B. E. Raines and T. Tennant, The specification property on a set-valued map and its inverse limit, arXiv: 1509.08415. |
[12] |
W. R. Utz,
Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774.
doi: 10.1090/S0002-9939-1950-0038022-3. |
[13] |
P. Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems (Proc. Conf. , North Dakota State Univ. , Fargo, N. D. , 1977), Lecture Notes in Math. , Springer, Berlin, 668 (1978), 231-244. |
[14] |
P. Walters,
Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78.
doi: 10.1016/0040-9383(70)90051-0. |
[15] |
R. K. Williams,
A note on expansive mappings, Proc. Amer. Math. Soc., 22 (1969), 145-147.
doi: 10.1090/S0002-9939-1969-0242143-4. |
show all references
References:
[1] |
N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Mathematical Library, 52. North-Holland Publishing Co. , Amsterdam, 1994. |
[2] |
J. -P. Aubin and H. Frankowska, Set-valued Analysis, Reprint of the 1990 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc. , Boston, MA, 2009. |
[3] |
R. Bowen,
ω-limit sets for axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.
doi: 10.1016/0022-0396(75)90065-0. |
[4] |
D. Carrasco-Olivera, A. R. Metzger and C. A. Morales, Logarithmic expansion, entropy and dimension for set-valued maps, Preprint, (2016), to appear. |
[5] |
D. Carrasco-Olivera, R. Metzger Alvan and C. A. Morales,
Topological entropy for set-valued maps, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3461-3474.
doi: 10.3934/dcdsb.2015.20.3461. |
[6] |
W. Cordeiro and M. J. Pacifico,
Continuum-wise expansiveness and specification for set-valued functions and topological entropy, Proc. Amer. Math. Soc., 144 (2016), 4261-4271.
doi: 10.1090/proc/13168. |
[7] |
M. Eisenberg,
Expansive transformation semigroups of endomorphisms, Fund. Math., 59 (1966), 313-321.
|
[8] |
J. P. Kelly and T. Tennant, Topological entropy for set-valued functions, arXiv: 1509.08413. |
[9] |
S. Y. Pilyugin and J. Rieger,
Shadowing and inverse shadowing in set-valued dynamical systems. Contractive case, Topol. Methods Nonlinear Anal., 32 (2008), 139-149.
|
[10] |
S. Y. Pilyugin,
Shadowing in Dynamical Systems Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999. |
[11] |
B. E. Raines and T. Tennant, The specification property on a set-valued map and its inverse limit, arXiv: 1509.08415. |
[12] |
W. R. Utz,
Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774.
doi: 10.1090/S0002-9939-1950-0038022-3. |
[13] |
P. Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems (Proc. Conf. , North Dakota State Univ. , Fargo, N. D. , 1977), Lecture Notes in Math. , Springer, Berlin, 668 (1978), 231-244. |
[14] |
P. Walters,
Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78.
doi: 10.1016/0040-9383(70)90051-0. |
[15] |
R. K. Williams,
A note on expansive mappings, Proc. Amer. Math. Soc., 22 (1969), 145-147.
doi: 10.1090/S0002-9939-1969-0242143-4. |
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